Open Ball in Euclidean Plane is Interior of Circle
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Theorem
Let $\R^2$ be the real number plane with the usual (Euclidean) metric.
Let $x = \tuple {x_1, x_2} \in \R^2$ be a point in $\R^2$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.
Then $\map {B_\epsilon} x$ is the interior of the circle whose center is $x$ and whose radius is $\epsilon$.
Proof
Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.
Let $y = \tuple {y_1, y_2} \in \map {B_\epsilon} x$.
Then:
\(\ds y\) | \(\in\) | \(\, \ds \map {B_\epsilon} x \, \) | \(\ds \) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \map d {y, x}\) | \(<\) | \(\, \ds \epsilon \, \) | \(\ds \) | Definition of Open $\epsilon$-Ball | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \sqrt {\paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2}\) | \(<\) | \(\, \ds \epsilon \, \) | \(\ds \) | Definition of Real Number Plane with Euclidean Metric | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2\) | \(<\) | \(\, \ds \epsilon^2 \, \) | \(\ds \) |
But from Equation of Circle:
- $\paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2 = \epsilon^2$
is the equation of a circle whose center is $\tuple {x_1, x_2}$ and whose radius is $\epsilon$.
The result follows by definition of interior and Open Ball of Point Inside Open Ball.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.2$